چكيده به لاتين
In the present thesis, we show how to apply both classical and nonclassical symmetry tech- niques to second-order linear Fokker-Planck equations (FPEs) of the form ut = −[f ′(x)u+f (x)ux]+ 1 2 g(t)uxx. This consists of determining the classical and nonclassical symmetries and then using the classical symmetries to simplify and resolve the governing equation; determining the optimal system of lie symmetry subalgebras; proposing general initial value problems (IVPs) that can be solved with the help of the lie symmetries (even those with initial conditions (ICs) that are not left invariant under the symmetries), and demonstrating how to apply symmetries to find functionally separable solutions. According to our nonclassical analysis: firstly, establish the relationship be- tween the arbitrary functions in the governing equation and the infinitesimals; then, based on the link, the system of determining equations of the nonclassical symmetry is expanded. New strictly nonclassical symmetries (NCLS) of the Fokker-Planck equation (FPE) in three main cases result, determined by assisting the Computer Algebra System Reduce. Furthermore, we explored using symmetries to solve initial value problems (IVPs). In general, the provided condition of the form u(x, 0) = F (x) is believed to require being left invariant under the one-parameter Lie group of transformations that leaves the partial differential equation (PDE) invariant in order to be able to solve IVPs. In accordance with this procedure, general ICs that can be solved by our governing FPE were built using the symmetries that were discovered. The method is explained via a number of examples. We further expanded the major outcome of the paper [1], allowing the most general form of a first-order IC to be permitted by a specific classical symmetry (CLS), by studying the work on symmetries and ICs. The generator was required to keep the initial condition’s invariance due to the results obtained in their study. This thesis extends their finding to the situation when the first-order IC is not required to be left-invariant, allowing for the solution of a large number of IVP using the symmetries. As a result, we offered some examples with classical symmetries (CLS) and set the outcome to our governing FPE with first-order ICs. Furthermore, based on the form of the invariant surface condition (ISC), we have demonstrated how to use the classical symmetries of FPE to find functionally separable solutions in different kinds of forms. To sum up, we have demonstrated in this thesis how classical symmetries approaches can be useful in locating solutions to FPEs, optimal systems, and IVPs. Furthermore, we have shown that functionally separable solutions can be found in a variety of forms by using the classical symmetries of FPEs.
